This chapter followed the learner: the child transcribing numbers, the pupil reading a word problem, the student writing a solution. The guide below gathers the habits that help at all these moments.

Six habits

  1. Read with a pencil in hand. A mathematics text is dense: one line may be worth a paragraph. Paul Halmos advised: “Don’t just read it; fight it!” Redo the calculations, invent examples, ask why each step holds.
  2. Read aloud. Saying $\sum_{k=1}^{n} k^2$ as “the sum of $k$ squared, for $k$ from one to $n$” forces you to recognise each part. Where speech stalls, reading stalled too.
  3. Look at the shape before calculating. What is the last operation? Are there repetitions, symmetries, a number that appears twice? Bogdanov-Belsky’s blackboard is solved by looking.
  4. Translate both ways. From text to formula, giving a name and a unit to each unknown; and from formula back to text, to check that it says what the problem says.
  5. Test with numbers. Before trusting a rule, substitute small values. A counterexample refutes a false rule; many examples do not prove a true one, but they give the confidence to look for a proof.
  6. Write for someone. Complete sentences, words between formulas, the goal announced before the calculations. If a classmate cannot follow the solution, it is not finished yet.

A complete reading of a badly written solution

A student hands in the following solution to “solve $2x + 3 = 11$”:

$$2x + 3 = 11 = 2x = 8 = x = 4.$$
  1. What is right: the answer, $x = 4$, and the calculations.
  2. What is written: a chain of equalities claiming $11 = 8 = 4$, which is false. The sign $=$ was used as “and then”, one of the uses that chapter 4 showed must be kept apart.
  3. The rule the student followed: “write what I do, in order, with equals signs between steps”, a consistent rule that mixes the register of speech with that of signs.
  4. A rewrite: “Subtracting $3$ from both sides, $2x = 8$. Dividing by $2$, $x = 4$. Checking: $2 \cdot 4 + 3 = 11$.”
  5. What was gained: each line is now true on its own, and the reader knows what was done between one and the next.

Glossary

  • Transcoding. Passing a number from one representation to another, such as from the spoken word to the written numeral.
  • Transcoding error. Writing a number the way it is spoken, piece by piece: $3004$ for “three hundred and four”.
  • Mathematics register. The variety of language used to talk about mathematics, with ordinary words in agreed meanings (Halliday).
  • Didactic contract. The set of unspoken rules that students and teachers follow without saying them, such as “every problem has an answer” (Brousseau).
  • Key word. A word in a problem taken as an order to perform an operation (“more” means add). It works sometimes and misleads often.
  • Convention. An agreed rule that could have been different: the order of operations, the decimal comma, the meaning of $\sin^2 x$.
  • False rule. A wrong rule built by analogy with a right one, such as $(a+b)^2 = a^2 + b^2$ from the distributive law (Matz).
  • Counterexample. A case in which a general claim fails. One is enough to refute it.
  • Procept. A symbol that is at once a process and the object it produces, such as $3+4$ or $13^2$ (Gray and Tall).
  • Instrumental and relational understanding. Knowing how to apply the rule, and knowing also why it holds (Skemp).
  • Linear notation. Writing on a single line, with opening and closing signs in place of position: Nemeth’s Braille, TeX, the calculator.
Exercise: apply the guide

A student writes: “$\sqrt{x^2 + 9} = x + 3$, so for $x = 4$ it gives $7$.” Use habit 5.

For $x = 4$: $\sqrt{16 + 9} = \sqrt{25} = 5$, not $7$. The rule used, $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$, is a false rule by analogy with the distributive law; a single test with numbers refutes it. The equality $\sqrt{x^2+9} = x + 3$ only holds for $x = 0$.

Exercise: read it aloud

Read aloud, without ambiguity: $\dfrac{1}{n}\displaystyle\sum_{k=1}^{n} x_k$. What does the expression compute?

“One over $n$, times the sum of $x$ sub $k$, for $k$ from one to $n$.” It is the arithmetic mean of the numbers $x_1, x_2, \ldots, x_n$: add them all and divide by how many there are. Saying the expression aloud helps to recognise a familiar idea in it.

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